Hecke algebras, modular categories and 3-manifolds quantum invariants

Abstract

We construct modular categories from Hecke algebras at roots of unity. For a special choice of the framing parameter, we recover the Reshetikhin-Turaev invariants of closed 3-manifolds constructed from the quantum groups U_q sl(N) by Reshetikhin-Turaev and Turaev-Wenzl, and from skein theory by Yokota. We then discuss the choice of the framing parameter. This leads, for any rank N and level K, to a modular category \tilde H^{N,K} and a reduced invariant \tilde\tau_{N,K}. If N and K are coprime, then this invariant coincides with the known PSU(N) invariant at level K. If gcd(N,K)=d>1, then we show that the reduced invariant admits spin or cohomological refinements, with a nice decomposition formula which extends a theorem of H. Murakami.